Theorem tospos 29658

Description: A Toset is a Poset. (Contributed by Thierry Arnoux, 20-Jan-2018.)

Assertion

Ref	Expression
tospos	|- ( F e. Toset -> F e. Poset )

Proof of Theorem tospos

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2622	. . 3 |- ( Base ` F ) = ( Base ` F )
2		eqid 2622	. . 3 |- ( le ` F ) = ( le ` F )
3	1, 2	istos 17035	. 2 |- ( F e. Toset <-> ( F e. Poset /\ A. x e. ( Base ` F ) A. y e. ( Base ` F ) ( x ( le ` F ) y \/ y ( le ` F ) x ) ) )
4	3	simplbi 476	1 |- ( F e. Toset -> F e. Poset )

Syntax hints:   -> wi 4   \/ wo 383   e. wcel 1990   A.wral 2912   class class class wbr 4653   ` cfv 5888   Basecbs 15857   lecple 15948   Posetcpo 16940  Tosetctos 17033

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-nul 4789

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-iota 5851  df-fv 5896  df-toset 17034

This theorem is referenced by:  resstos  29660  tltnle  29662  odutos  29663  tlt3  29665  xrsclat  29680  omndadd2d  29708  omndadd2rd  29709  omndmul2  29712  omndmul  29714  isarchi3  29741  archirngz  29743  archiabllem1a  29745  archiabllem2c  29749  gsumle  29779  orngsqr  29804  ofldchr  29814  ordtrest2NEWlem  29968  ordtrest2NEW  29969  ordtconnlem1  29970